scholarly journals An Approximate Analytical Approach for Systems of Fredholm Integro-Differential Equations of Fractional Order

2021 ◽  
Vol 15 ◽  
pp. 91-104
Author(s):  
Mudaffer Alnobani ◽  
Omar Abu Al Yaqin
Keyword(s):  
Differential Equations ◽  
High Accuracy ◽  
Ease Of Use ◽  
New Technique ◽  
Computational Time ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
A New Technique ◽  
Residual Power

A new technique for solving a system of fractional Fredholm integro-differential equations (IDEs) is introduced in this manuscript. Furthermore, we present a review for the derivation of the residual power series method (RPSM) to solve fractional Fredholm IDEs in the paper done by Syam, as well as, corrections to the examples mentioned in that paper. The numerical results demonstrated the new technique’s applicability, efficacy, and high accuracy in dealing with these systems. On the other hand, a comparison has been done between the two schemes using the two corrected examples in addition to a problem that had been solved in many previous studies, and the results of these studies were compared with the new technique and RPSM. The comparison demonstrated clear superiority of our method over the RPSM for solving this class of equations. Moreover, they dispel the misconception that the RPSM works effectively on fractional Fredholm IDEs as mentioned in the paper done by Syam, whereas two problems solved by the RPSM produced an unaccepted error. Also, the comparison with the previous studies indicates the importance of the new method in dealing with the fractional Fredholm IDEs despite its simplicity, ease of use, and negligible computational time.

scholarly journals Applications of General Residual Power Series Method to Differential Equations with Variable Coefficients

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Bochao Chen ◽  
Li Qin ◽  
Fei Xu ◽  
Jian Zu
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Variable Coefficients ◽  
Series Solutions ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Analytical Series ◽  
Residual Power

This paper is devoted to studying the analytical series solutions for the differential equations with variable coefficients. By a general residual power series method, we construct the approximate analytical series solutions for differential equations with variable coefficients, including nonhomogeneous parabolic equations, fractional heat equations in 2D, and fractional wave equations in 3D. These applications show that residual power series method is a simple, effective, and powerful method for seeking analytical series solutions of differential equations (especially for fractional differential equations) with variable coefficients.

scholarly journals New Implementation of Residual Power Series for Solving Fuzzy Fractional Riccati Equation

2018 ◽  
Vol 10 (2) ◽  
pp. 81
Author(s):  
Moath Ali Alshorman ◽  
Nurnadiah Zamri ◽  
Mohammed Ali ◽  
Asia Khalaf Albzeirat
Keyword(s):  
Power Series ◽  
Riccati Equation ◽  
Fractional Power ◽  
High Accuracy ◽  
Computational Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Fractional Power Series ◽  
Residual Power

This paper reveals a computational method using a Residual Power Series Method (RPSM) for the solution of fuzzy fractional riccati equation under caputo fractional differentiability. An analytical solution of fuzzy fractional riccati equation is obtained as a convergent fractional power series. The procedure produces solutions of high accuracy, and some illustrative examples are solved with a different value of orders to show the efficiency of the RPSM.

scholarly journals Toward computational algorithm for time-fractional Fokker–Planck models

2019 ◽  
Vol 11 (10) ◽  
pp. 168781401988103 ◽  
Author(s):  
Asad Freihet ◽  
Shatha Hasan ◽  
Mohammad Alaroud ◽  
Mohammed Al-Smadi ◽  
Rokiah Rozita Ahmad ◽  
...  
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Power ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Wide Range ◽  
Residual Power Series ◽  
Residual Power ◽  
Fokker Planck

This article describes an efficient algorithm based on residual power series to approximate the solution of a class of partial differential equations of time-fractional Fokker–Planck model. The fractional derivative is assumed in the Caputo sense. The proposed algorithm gives the solution in a form of rapidly convergent fractional power series with easily computable coefficients. It does not require linearization, discretization, or small perturbation. To test simplicity, potentiality, and practical usefulness of the proposed algorithm, illustrative examples are provided. The approximate solutions of time-fractional Fokker–Planck equations are obtained by the residual power series method are compared with those obtained by other existing methods. The present results and graphics reveal the ability of residual power series method to deal with a wide range of partial fractional differential equations emerging in the modeling of physical phenomena of science and engineering.

scholarly journals Numerical Investigation to Fuzzy Volterra Integro-Differential Equations via Residual Power Series Method

2020 ◽  
pp. 1-7
Author(s):  
Mohammad Alshammari ◽  
Mohammed Al-Smadi ◽  
Ishak Hashim ◽  
Mohd Almie Alias
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Effective Technique ◽  
Power Series Method ◽  
Solution Approach ◽  
Residual Power Series Method ◽  
Residual Functions ◽  
Residual Power Series ◽  
Residual Power ◽  
Taylor’S Series

In this paper, a study of a numerical approximate solution to fuzzy Volterra integro-differential equations is presented under strongly generalised differentiability by applying an influent effective technique, called the Residual Power Series (RPS) method. The solution approach can be expressed on Taylor's series formula in terms of elementary σ-level representation, whereas the coefficients can be computed by utilising its residual functions. Furthermore, a numerical computational example is given to test and validate the proposed method. The results reached show several features concerning the RPS method such as potentiality, generality and superiority to handle many problems arising in physics and engineering.

scholarly journals Numerical Investigation to Fuzzy Volterra Integro-Differential Equations via Residual Power Series Method

2020 ◽  
pp. 1-7
Author(s):  
Mohammad Alshammari ◽  
Mohammed Al-Smadi ◽  
Ishak Hashim ◽  
Mohd Almie Alias
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Effective Technique ◽  
Power Series Method ◽  
Solution Approach ◽  
Residual Power Series Method ◽  
Residual Functions ◽  
Residual Power Series ◽  
Residual Power ◽  
Taylor’S Series

In this paper, a study of a numerical approximate solution to fuzzy Volterra integro-differential equations is presented under strongly generalised differentiability by applying an influent effective technique, called the Residual Power Series (RPS) method. The solution approach can be expressed on Taylor's series formula in terms of elementary σ-level representation, whereas the coefficients can be computed by utilising its residual functions. Furthermore, a numerical computational example is given to test and validate the proposed method. The results reached show several features concerning the RPS method such as potentiality, generality and superiority to handle many problems arising in physics and engineering.

Adaptation of residual power series method to solve Fredholm fuzzy integro-differential equations

2019 ◽  
Author(s):  
Mohammad Al Shammari ◽  
Mohammed Al-Smadi ◽  
Omar Abu Arqub ◽  
Ishak Hashim ◽  
Mohd Almie Alias
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Residual Power

scholarly journals Numerical Investigation to Fuzzy Volterra Integro-Differential Equations via Residual Power Series Method

2020 ◽  
Vol 13 ◽  
pp. 1-7
Author(s):  
Mohammad Alshammari
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Effective Technique ◽  
Power Series Method ◽  
Solution Approach ◽  
Residual Power Series Method ◽  
Residual Functions ◽  
Residual Power Series ◽  
Residual Power ◽  
Taylor’S Series

In this paper, a study of a numerical approximate solution to fuzzy Volterra integro-differential equations is presented under strongly generalised differentiability by applying an influent effective technique, called the Residual Power Series (RPS) method. The solution approach can be expressed on Taylor's series formula in terms of elementary σ-level representation, whereas the coefficients can be computed by utilising its residual functions. Furthermore, a numerical computational example is given to test and validate the proposed method. The results reached show several features concerning the RPS method such as potentiality, generality and superiority to handle many problems arising in physics and engineering.

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